Resaerch On Hardy Type Inequalities With Remainder Terms

The ideas and techniques of Harmonic Analysis have penetrated into almost all areas of mathematics, especially it is closely related to the research of differential equations. Hardy inequalities have special roles in a number of researches in partial differential equations, particularly the sharp Hardy inequality is useful in handling with the existence, regularity, eigenvalue problem of elliptic equations with singular low order term. The researches on various Hardy type inequalities with best constant, by using precise tools and methods of harmonic analysis, have caused widespread concern in the mathematical community at home and abroad.The overall objective of this paper is based on available information and the trend of latest research at home and abroad, using and developing the accumulated research methods and techniques in harmonic analysis and partial differential equation, systematically studying the theory and method related to Hardy inequalities, exploring the fields of little previous study with broad idea, and striving to achieve for extensive application of the results.The main content of this article contains the following three aspects:(1) Prove two improved Hardy type inequalities with remainder terms in Euclidean space, and get the best constant and considered the conditions of reaching the best constant.(2) Prove a kind of Hardy type inequality with remainder terms in the anisotropic Heisenberg group.(3) Prove a kind of Hardy inequality with remainder terms corresponding to anisotropic Greiner vector fields.Although many domestic and foreign scholars have done a lot of research on the Hardy inequalities, academic literature on Hardy inequalities with remainder terms is relatively less. The result of this paper contains the latest achievements and promotion in this field, by taking some new ways to get the desired results, and explores some of the more cutting-edge contents.